On the Rigidity of Hamiltonians which are Zoll Near a Minimum, with an Application to Magnetic Systems and Almost-K\"ahler Manifolds

Abstract

We study Hamiltonian systems near a compact symplectic Morse-Bott minimum. Our first result shows that if the flow is Zoll (that is, it induces a free circle action) along a sequence of energy levels converging to the minimum, then the Hessian of the Hamiltonian in the symplectic normal directions must be compatible with the restriction of the symplectic structure to the normal bundle (that is, its representing endomorphism is a complex structure of the symplectic normal bundle). For our second result, we specialize to magnetic systems on closed manifolds with symplectic magnetic form. In this setting, if the system is Zoll along a sequence of energy levels converging to the minimum, then the metric is compatible with the magnetic form and therefore defines an almost K\"ahler structure. We show that a natural curvature quantity, consisting of the holomorphic sectional curvature corrected by a term measuring the non-integrability of the almost complex structure, must be constant. In particular, we obtain a dynamical characterization of complex space forms among K\"ahler manifolds. Together, these results establish strong rigidity of systems which are Zoll at energies close to a Morse-Bott minimum, in the symplectic and in the magnetic settings.